Please feel free to reconsider my application in view of my addressing to the reviewers concerns. Also, please feel free to ask questions. This theory is not written in the language standard scientists are familiar with.

Thanks,

Marco Pereira

Reviewer 1

Dear Editor,

The author seeks to interpret the data for the dependence of luminosity of supernovae on redshift as evidence for a rather revolutionary cosmological model, the Hypergeometric Universe (HU), claiming that the fit is at least as good as the concordance model of cosmology.

I must admit that I am very confused by what the author proposes. It appears to me that, at least initially, the HU does not change any of the standard laws of physics but only changes the geometry in which the Universe is embedded. He proposes that the observed universe is the surface of a 3-sphere expanding into a new non-compact spatial direction. If this indeed the fundamental proposition, then it does not work:

Models of this type were studied extensively in the late 90s/early 2000s under the guise of braneworlds (see e.g. the Randall-Sundrum model), but the extra dimensions were compact in order to prevent the existence of low-mass Kaluza-Klein modes. The size of this extra dimension would have to be microscopic, of the order of inverse TeV in order to avoid this problem.

* Even if one were to brush this off as some sort of new physics (e.g. inspired by open string theory), gravity must know about the extra dimensions. Again, we know with excellent precision that the force of gravity in the solar system falls off as 1/r^2 and therefore it is 3+1 dimensional. It is possible to have a model of gravity which is 4D at small distances but sensitive to the full 5D space at large distances (see the Dvali-Gabadadze-Porrati model) but this means that the universe behaves in a completely standard way until the acceleration era. This is not what the author has in mind and comes with its own inconsistencies (ghosts etc) which have not been solved.

* The recent detection of gravitational waves by LIGO confirms that we understand reasonably well the generation and propagation of these waves on cosmological distances. For the reason explained above, they would also have been sensitive to the full 5D structure of the space time and therefore their luminosity would have been completely different in such a 5D setup.

* I do not understand why the author uses the standard equation for luminosity distance in LambdaCDM cosmology (calling it the "improved Hubble law"). This fundamentally assume an FRW geometry in 3 spatial dimensions, with the cosmological constant and non-relativitistic matter as the only sources of energy in the universe. If the author would like to make claims about how well supernovae fit his model, he should start with a metric and with a definition of how light propagates (e.g. on null geodesics in standard physics). Given the metric and its dynamics which one would obtain by solving Einstein equations, an equation for th luminosity distance-redshift relation can be derived. It would be different.

The rest of the paper in fact makes use of some simple geometrical arguments to obtain distances and I have to admit I do not understand what they refer to. Proper null geodesics of the metric must be calculated to be able to say anything about distances.

* I do not understand the assertion that the speed of light is \sqrt{2}c. What is light for the author? What is the experiment which would give such a result?

* Finally, I do not think it is possible for G to vary in the way that the author proposes. For one, we have constraints on the variation of G on Earth from the Oklo natural nuclear reactor and, going back further into the past, from Big Bang Nucleosynthesis which mean that if cannot have change by more than a few percent. If it were changing together with Hubble, we would find that the galaxies and orbits destabilise etc. Moreover, as the author notices, stars are very sensitive to the strength of gravity (e.g. arXiv:1102.5278) and we would have know about such variations.

So overall, I am not convinces that the author has properly calculated the actual impact of the geometry of the universe that he proposes. Since he is not modifying physics in a fundamental way, but only the geometry (which is arguably the appeal of his model), many experiments in physics can be reinterpreted in terms of this new setup. As far as I can tell, they would not allow the set up to be possible. Thus unless the author can convincingly prove that the standard well known local physics is not modified in his setup, it is premature to try to calculate the impact on cosmology.

Moreover, I should say that today the geometry of the universe is much more precisely constrained by angular diameter distance measurements of the baryon acoustic oscillations in galaxy correlation functions. This agrees very well with the supernova data and I suspect very much would be a very different prediction in the HU model.

This manuscript presents the predicted luminosity distance in the Hypergeometrical Universe (HU) model and compares against the Union supernova compilation. Unfortunately, I do not believe that it is suitable for publication. I detail (only the major) issues below.

The gravitational constant changing dramatically throughout the history of the universe is disfavored by growth-of-structure constraints, pulsar-timing experiments, Solar-system tests (e.g., Lunar distance), stellar evolution, and so forth.

I’m also skeptical that the luminosity of a SN Ia if G were different would scale as G^-3 (or M_ch^2). Ni-56 production is not a simple rate-limited process; SNe Ia undergo a deflagration that (in most cases) transitions to a detonation. They burn about half their mass to Ni-56 (depending on when the detonation occurs). Even if Ni-56 production were a simple process, the radius (and thus the density) of the white dwarf also changes with G.

But putting all of that aside, I will take a narrow view of the manuscript. It proposes a distance(redshift) relation, and we can quantitatively see how well this matches the data. The proper way to do this is not by making plots, it is to compute chi^2 values from the distance moduli (mu) and covariance matrix in Union2.1:

where M is a constant that can be fit (the host-mass relation can also be fit, but failing to do so won’t affect the results much). After computing chi^2 values for LambdaCDM and HU, you can see if HU is favored or disfavored by the data compared to LambdaCDM. By my eye, HU is significantly worse, but the chi^2 values will say for sure.